Damping modelling

Damping modelling

• Viscous damping, E_v = c · Δx
• Hysteretic or structural damping
• Coulomb friction (dry friction contact)

4. Viscous

C = viscous damping coeff. ξ = damping factor.

Estimation methods: peak of frequency response function (freq. domain approach); from free response (time response approach).

i = 1.

t_i, t_i+1 → x₁, x₂.

t_i+1 = t_i + T_d = t_i + 2π/ω_d.

x(t_i) = x₁ = C · e^-ξω_nt_icos(ω_dt_i - φ).

x(t_i+1) = x₂ = C · e^-ξω_nt_i+1cos(ω_dt_i+1 - φ).

x₁/x₂ = e^2πξ/N=LN=x.

ln x₁/x₂ = δ = 2πξ/√(1 - ξ²).

ξ = δ/^√2π2 + δ².

if ξ << 1 → δ = 2πξ → ξ = δ/2π.

Damping modelling

• Viscous
• Hysteretic or structural damping
• Coulomb friction (dry friction contact)
• Viscous

C: viscous damping coeff.

ξ: damping factor.

Estimation methods: peak of frequency response function (freq. domain approach); from free response (time response approach).

t₁, t₂ → x₁, x₂.

t₂ = t₁ + T_d - t₁ = 2π / ω_d.

x(t₁) / x(t₂) = x₁ / x₂ = C · e^σt₁.

ln x₁ / x₂ = δ = -2πξ / √(1-ξ²).

ξ = δ / √(2π² + δ²).

if ξ << 1 → δ ≈ 2πξ → ξ ≈ δ / 2π.

Damping modelling

• Viscous: Fv = c. Δẋ
• Hysteretic or structural damping
• Coulomb friction (dry friction contact)

(1) Viscous

c: viscous damping coeff.

Ϛ: damping factor.

Estimation methods: peak of frequency response function (freq. domain approach); from free response (time response approach).

μ = 1.

t₁, t₂ → x₁, x₂.

t₂ = t₁ + T₀ → t₁ + 2π_ω.

x(t₁)/x(t₂) = x₁/x₂ = C₁.e^-Ϛωₙt₁cos(ω_dt₁ - φ) / C₁.e^-Ϛωₙt₂cos(ω_dt₂ - φ).

x₁/x₂ = e^2πϚ/N=Ϛ.

ln x₁/x₂ = δ = 2πϚ / √(1-Ϛ²).

Logarithmic decrement.

Ϛ = δ/√(2π)² + δ².

if Ϛ << 1 → δ = 2πϚ → Ϛ = δ/2π.

For more accurate estimates, more than 2 peaks: x₁, x₂, ..., x_i+2.

S = ¹/_i ln ^x₁/_xi+2.

More accurate.

We are averaging the response constant on longer part of the response essentially as we did to do when we considered 2 consecutive peaks.

2) Hysteretic damping

(Structural).

Dissipation for each cycle.

W_h = ∫ f_h dx = ∫ cẋ² dx =.

Solution under harmonic motion.

Viscous damping.

Energy dissipated by viscous damper at each loading cycle.

Hysteretic dampers.

W_h = αx₀²~> Experimental.

Equivalence in terms of dissipated energy:

W_v = W_h => ^{C_eq = α}/_ωx0 = ^k/_ω.

(The slope of the curve depends on k).

Structural dampers depend on ω.

c) Study the steady state response for harmonic excitation

mẍ + C_eqẋ + kx = f(t) = F₀e^iωt.

mẍ + ^h/_ωiωx + kx = F₀e^iωt.

mẍ + (K + iωh)x = F₀e^iωt.

K_c = K(1+iM).

Complex stiffness; η = _h/^K loss factor.

x(t) = _F0/^K e^iωt.

The reectance freq. resp. function.

H(ω) = _x0/^F₀/k = ¹/_{1+iM - ^ω2/ωn²ⁱ} = ^{1 - ω2/_ωn2 - iM}.

|H(ω)| = ¹/_{√((1 - ^ω2/ωn²)² + M²)}.

ψ = arctan ( ^M/_{1- ^ω2/ωn²}).

|H|1ω/ω_nψ.

G(0,1) graph is not valid in this case for small frequencies because there is no more for small freq. the meaning of G for static response.

• if ω → 0 |H(ω)|= ¹/_√(1+M²) < 1

if ψ = arctan M ≠ 0.

• ω = ω_n

|H|_max = ¹/_M.

eq. with viscous |H|_max = ¹/_2ζeq.

if ζ ≡ M/2.

for underdamped system → ¹/_{√(1-c) < √2} 0 < c < √2.

• Hysteretic damping is only valid in steady-state oscillatory motion
• Not suitable for initial vib...

Energy loss per cycle is independent of speed (ẋ) and.

3. Coulomb friction

Friction force: _Ff = µN = µmg.

_Fd = _Ff sign (ẋ) = µN sign (ẋ).

EOM.

mẍ + kx ± _Fd sign (ẋ) = 0.

Not linear (system goes back and forth) but we can write a linear EOM:

mẍ + kx = -_Fd, ẋ ≥ 0.

mẍ + kx = _Fd, ẋ < 0.

Study the free response from i.c. x(0) = x_o, ẋ(0) = 0.

Motion starts if: |_kxo| > 1 / µmg.

• ẍ + ω_n²x = ω_n² _Ff/k, x < 0
• ẍ + ω_n²x = -ω_n² _Ff/k, ẋ > 0

Which eq.?

If x_o > 0, mass begins to move to the left ẋ < 0 ⟹ eq. 1.

x(t) = _Ff/k + a cos (ω_nt) + b sin (ω_nt).